Parallel Vectors & Unit Vectors (Edexcel IGCSE Maths B): Revision Note

Parallel vectors

How can I tell if two vectors are parallel?

• Two vectors are parallel if one is a scalar multiple of the other

  • This means that all components of the vector have been multiplied by a common constant (scalar)

  • For example, $\left(\begin{array}{c}1\\ 3\end{array}\right)$ and $\left(\begin{array}{c}2\\ 6\end{array}\right)$ are scalar multiples

    • The numbers in the first vector have each been multiplied by 2 to get the numbers in the second vector

• Multiplying the components of a vector by a positive scalar changes the magnitude of the vector but not the direction

  • For example, $\left(\begin{array}{c}2\\ 6\end{array}\right)$ is double the length of $\left(\begin{array}{c}1\\ 3\end{array}\right)$ but in the same direction

• Multiplying the components of a vector by a negative scalar reverses the direction

• You can factorise a vector to help spot if two vectors are parallel

  • $\left(\begin{array}{c}9\\ -3\end{array}\right)=3\left(\begin{array}{c}3\\ -1\end{array}\right)$

  • $\left(\begin{array}{c}-6\\ 2\end{array}\right)=-2\left(\begin{array}{c}3\\ -1\end{array}\right)$

    • They are scalar multiples of the same vector so they are parallel

Unit vectors

What is a unit vector?

• A unit vector is a vector with a modulus (length) of 1

• To find a unit vector that is in the same direction as the vector $\mathbf{a}$

  • divide the components of the vector by the modulus of the vector

  • I.e. $\frac{\mathbf{a}}{\left|\mathbf{a}\right|}$ is a unit vector in the direction of vector $\mathbf{a}$

• For example, a unit vector in the direction $\left(\begin{array}{c}3\\ -4\end{array}\right)$ is

$\begin{array}{rcl}\frac{1}{\sqrt{3^2+{\left(-4\right)}^2}}\left(\begin{array}{c}3\\ -4\end{array}\right)& =& \frac{1}{5}\left(\begin{array}{c}3\\ -4\end{array}\right)\\ & =& \left(\begin{array}{c}\frac{3}{5}\\ -\frac{4}{5}\end{array}\right)\end{array}$